PHYSICAL REVIEW E JUNE 1999VOLUME 59, NUMBER 6

Anomalous transport and quantum-classical correspondence

Bala Sundaram1 and G. M. Zaslavsky21Department of Mathematics, CSI-CUNY, Staten Island, New York 10314

2Courant Institute of Mathematical Sciences, New York University, New York, New York 10012and Department of Physics, New York University, 2-4 Washington Place, New York, New York 10003

~Received 25 February 1998; revised manuscript received 3 March 1999!

We present evidence that anomalous transport in the classical standard map results in strong enhancement offluctuations in the localization length of quasienergy states in the corresponding quantum dynamics. Thisgeneric effect occurs even far from the semiclassical limit and reflects the interplay of local and globalquantum suppression mechanisms of classically chaotic dynamics. Possible experimental scenarios are alsodiscussed.@S1063-651X~99!03006-8#

PACS number~s!: 05.45.2a, 03.65.Sq, 05.60.2k

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It is generally accepted that quantum mechanics spresses chaotic classical motion. Numerous studies hidentified mechanisms for suppression which, for our pposes, fall into two broad classes. One class is exemplby ‘‘dynamical localization’’ where quantum eigenstates alocalized in a variable such as momentum despite the deministic diffusion seen in the limiting classical dynamics@1#.This effect is analogous to Anderson localization in tigbinding models@2# and, in one-dimension, the localizewave function has a characteristic exponential form. Tscale is the localization lengthj which is related to the classical diffusion constant, an important fact for the work reported here. There is also a time scalet* beyond whichquantum and classical dynamics deviate. Despite a few coterexamples, dynamical localization provides a ‘‘globamechanism for quantum suppression.

The second class can be motivated by the fast deviatiowave packet dynamics from the classical motion even insemiclassical limit. This effect is contained in the logaritmic break timet\5 ln(I/\)/G for the breakdown of quantumclassical correspondence@3#, relative to a characteristicLyapunov exponentG and actionI. However, the quantumdynamics retains features of the classical dynamics for timbeyond this estimate. ‘‘Scarring’’@4,5# refers to quantumcoherences associated with local, unstable and marginstable, classical invariant structures. Though many oquestions remain, this effect has been numerically andperimentally observed in a variety of strongly coupled qutum systems.

We present evidence of another example of ‘‘classipersistence’’ due to anomalous diffusion and acceleramodes in chaotic dynamics. There has been considerabltivity in the area of anomalous classical transport@6–8#. Ofparticular relevance to the present work is that even inestablished paradigm such as the standard map, describqn115qn1pn11 and pn115pn1K sinqn , surprising newresults were reported.

In the limit of largeK, the classical standard map dynamics is diffusive in the action variablep with diffusion con-stant ^p2&/t5K2/2[Dql . What is also established is thaaway from this limit, the diffusion constant exhibits peawith changingK and varies in time@8#. Recent results@6,7#with improved precision and detail show that there are val

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of K where the effective diffusion constant~over some time!can be different fromDql by many orders of magnitudeFigure 1 shows these windows of anomalous diffusion whare not very narrow and exhibit substructure in each ofpeaks.

It was shown that there is no diffusion at all and that trandom walk process corresponds to Le´vy-like wanderingwith a transport exponentm given by ^p2&'tm, where m.1. m varies withK and has the ‘‘normal’’ valuem51 onlyat special values ofK. Superdiffusion occurs form.1 lead-ing to anomalous growth in momentum. It was explicitshown@6# that this could result from an island hierarchy wia peculiar topological structure near ‘‘accelerator’’ modwhich appear at special values ofK. Near these values, classical trajectories stick to the boundaries of these islawhich leads to flights of arbitrary length. The net resultstrong intermittency and superdiffusion.

One signature of anomalous transport is constructed fthe set of recurrence times$t j%, j 50,1, . . . , for aclassicaltrajectory originating in a region of phase space. The pr

FIG. 1. Diffusion coefficientD for the standard map, normalizeto Dql5K2/2, as a function ofK. The deviations from the quasilinear prediction span a wide window ofK and, as seen from the insecan be large at select values ofK, even far from the bifurcationpoint K52p.

7231 ©1999 The American Physical Society

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ability distribution R(t) of Poincare´ cycles $t j%5$t j 112t j% can now be computed. For ‘‘perfect’’ mixing and nomal diffusion,R(t) is strictly Poissonian whereas it exhibipowerlike asymptotics for anomalous kinetics. This classcharacteristic of anomalous transport is illustrated in Figand is important for the corresponding quantum dynamicwell.

We pose several questions, arising from classical anolous diffusion, in the corresponding quantum dynamicsthe standard map, or, equivalently, the ‘‘delta-kicked rotoH5p2/21K cosq(nd(t2n). The scale lengthj is related tothe classical diffusion constant and even follows the vations in D with K @11#. Thus, we anticipate that strong ehancement inD resulting from anomalous diffusion shoualso be reflected inj. The nature of dynamical localization ithese special windows is a related question. Further, as slocal structures in the classical phase space lead to the sdiffusion, these issues directly pertain to the interplay ofcal and global aspects of quantum suppression. This idereinforced by the enhanced diffusion that ought to be visiin the quantum dynamics even far from the semiclasslimit, where scarring is significant. Other authors have csidered quantum dynamics in the presence of anomatransport@9,10# though in a different regime. We focus othe situation where the quantization scale 2p\ is muchlarger than the size of the islands. Thus, any coherencesociated with the islands would be due to ‘‘scarred’’ quatum states.

Consider the evolution of̂p2& with time t ~measured asthe number of kicks! in the quantum dynamics for two vaues of the parameterK. The computation uses fast-Fouritransforms to evolve a plane wave initial state under repeapplication of the single-kick evolution operator

U5exp~2 ip2/2\!exp@2 iK cos~q!/\#. ~1!

The expectation values are then computed from the tievolved wave function.Kc56.908745••• is a critical value@6# in a wide parametric window dominated by anomalotransport. We illustrate this regime by consideringK*56.905 and contrast it withK511, where the effective dif-

FIG. 2. Distribution of Poincare´ cycles. The inset clearly displays the power-law tail of the distribution.

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fusion coefficient is actually belowDql ~seen in Fig. 1!. \52pa, where several irrational values ofa in the range0.0421 were considered. In the quasilinear limit, the quatum diffusion is expected to saturate att* 5aDql /\

2 wherea was numerically shown to be 1/2@1,11#. j measured inangular momentum quanta is't* . Figure 3 shows^p2&/(Dqlt* ) as a function of time for two representativvalues ofa. For largea, both values ofK result in strongsaturation withK511 having a higher saturation value thaK* 56.905. However, with decreasinga the enhanced diffu-sion for K* 56.905 becomes evident and there is a slodown in growth in lieu of saturation~over the time consid-ered!. By contrast,K511 appears to saturate at a valwhich is about an order of magnitude lower. This differenis considerably larger than estimates based on quasilianalysis. The inset in Fig. 2 verifies the existence of powlaw behavior in^p2& for longer timesat K* as comparedwith saturation atK511. Note that the slope points to subdiffusive rather than to superdiffusive growth for which whave no clear explanation at present.

A stronger signature appears on considering the variaof localization lengthj with K. It was shown@11# that jtracks the variations in the classical diffusion coefficientDwhen plotted with respect toKq5K sin(\/2)/(\/2), an im-portant correction at larger\. Figure 4 shows the variation inj, normalized to the quasilinear estimate, with respect toKq .j is obtained by fitting an exponential to the time-dependwave function for long times. The peaks in the inset coincwith the classical accelerator modes atK52p and 4p andjcomputed at different times reflect the same features.detailed scan displays a reasonable correspondence wit

FIG. 3. Variation of^p2& ~normalized to the saturation valuDqlt* ) with the number of kickst. Two representative values ofa,one large (Q,a'0.61803399) and the other small (SC,a'0.042340526), are shown forK511 ~dashed lines! and K*56.905 ~solid lines!. Note that the largera value was time-averaged to remove fluctuations that mask the trends. The ishows a log-log plot~at longer times,'5000 kicks! for a50.042340526,K* 56.905 ~upper curve!, and K511.0 ~lowercurve!. The corresponding slopes are 0.21~solid line! and 0.06~dashed line! respectively.

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results shown in Fig. 1, despite being far from the semicsical limit. It should be noted that the highest value occursKq'Kc defined earlier.

We now consider the localized wave function when tclassical transport is predominantly anomalous. Figureshows the evolved momentum distribution averaged o600 kicks. Panel~a! corresponds toKq at a peak in Fig. 4while ~b! sits in the valley. Localization does occur thoug

FIG. 4. Oscillations in the localization lengthj normalized tothe quasilinear estimate as a function ofKq . a'0.131267463which makes the corrections to the classical stochasticity paramK significant. The inset shows peaks associated with the accelemodes while the main figure zooms in on the region considereFig. 1. The inset also showsj computed after 3000 kicks~points! aswell as 9000~line! kicks. Note that the wave function averaged ov600 kicks is used to determinej.

FIG. 5. Line shapes after 6000 kicks starting from a plane winitial condition.~a! corresponds toKq'Kc56.90845••• while ~b!corresponds toKq510.69. a'0.131267463 and the noise in thline shapes was reduced by averaging over 600 kicks.

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the characteristic line shape is strongly nonexponentialKq in the neighborhood of the peaks. The contrast withclearly exponential line shape shown in~b! is striking. Thecentral region in~a! also displays curvature and an exponetial fit yields aj which is considerably larger than the qusilinear prediction. However, in case~b!, the computedj iswell approximated by the quasilinear estimate. Moving awfrom the peak inKq decreases the prominence of theshoul-ders in the line shape. The shoulders develop over a ti~which depends on\) which is consistent with the crossovetime from exponential to power-law behavior in Fig. 1.

Arguments for deviations of the localization length frothe simple estimates proceed from the long-time averamomentum distribution̂ f n&5(mucm(n0)u2ucm(n)u2, start-ing from a plane waven5n0 initial condition.cm(n) is theplane wave representation of the eigenfunction with quasergy vm . The analysis of localization assumes thatquasienergy states are exponentially localized and thatfluctuations inj are small. If this were true in the case oanomalous transport, the oscillations in Fig. 4 should direccorrelate with the number of quasienergy states participain the dynamics. To address this view, we computedparticipation ratioPr from the time averaged autocorrelatiofunction

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As seen from Fig. 6, the only correlation with Fig. 4 occuin the valley where the line shape is exponentially localizAt the peaks, these results strongly support the ideaanomalous transport results in strong fluctuations inj amongthe individual quasienergy states. This was verified by cstructing the ‘‘entire’’ spectrum ofU under conditions nec-essary for dynamical localization. The average value oj

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FIG. 6. Participation ratio as a function ofKq . Time averages ofthe autocorrelation function over 1000~points! and 6000~line! kicksare shown.a'0.131267463.

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does not reflect the large changes inD though the dispersionDj exhibits large fluctuations. This view is being explorfurther at present.

Theoretically, the classical flights of arbitrary length rsult ~for the standard map! from a hierarchy of boundarylayer islands@6#. The island areaSn , associated periodTn

and Lyapunov exponentGn scale asSn5lSnS0 , Tn5lT

nT0,and Gn5lG

nG0 with the generationn of the island chain.lS,1, lT.1, andlG'1/lT .

Quantum flights are possible as long asSn>\ leading toa critical valuen\5u ln \/S0u/uln lSu. The corresponding timeis T\5T0(S0 /\)1/m, wherem5u ln lSu/ln lTu.1 is a classicaltransport exponent@6#. We generalize the break timet\ de-fined earlier to ln(I/\)/Gn where I was some characteristiaction. The longest time corresponds to the smallestGn5G\51/T\ from which

t\5T\ ln~ I /\!5T0~S0 /\!1/m ln~ I /\!. ~3!

The result points to power-law dependence on\ as well as apossible crossover in scaling behavior. Note thatT\ forflights is a pure quantum effect, necessary for localizatiwhich cannot occur if arbitrarily long flights exist with hig

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probability ~not exponentially small!. Theoretical estimatesare considerably harder when\.S0.

The issue of fluctuations inj is relevant to an atom opticrealization of the kicked pendulum@12#. Here, the localiza-tion properties are similar to those in the quantum standmap, though small islands persist in the classical dynamAs we have illustrated, this could lead to strong fluctuatioin the localized wave function. More recent experimenhave realized thed-kicked rotor system@13# where the mea-surement of diffusion, over time scales longer than thopredicted earlier, can provide direct evidence of any anomlous transport. We note that signatures of anomalous traport are similar to those resulting from external noise. Tmay be especially relevant in the weak noise limit and wbe considered in detail elsewhere.

The work of B.S. was supported by the National ScienFoundation and a grant from the City University of NeYork PSC-CUNY Research Award Program. G.M.Z. wsupported by U.S. Department of Navy Grants Nos. N00093-1-0218 and N00014-97-1-0426. The authors are gratto Mark Edelman for his assistance with the Poincare´ recur-rence plot.

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